Choices for A in the Matrix Equation T=AB-BA

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Abstract
If F is any field and T is an n × ntrace zero matrix over F, then T= AB - BA where A and B are n × n matrices over F and A is nonsingular. If F has n distinct elements and T is not a multiple of In, the identity matrix, then A may be chosen to have n distinct characteristic roots. If F is algebraically closed, then in addition the characteristic roots of B may be arbitrarily prescribed. However, if n = AB - BA, then F has characteristic p > 0 and p divides the multiplicity of each of the characteristic roots of A and of B. Let adA(B) = AB — BA.lf F has n distinct elements and T is not a multiple of In , then for each integer m there exist A and B such that [ILM0001]. However, if T=In where n = prk then such A and B exist if and only if m < pr .

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